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Winter 2002 Notes on the existence of certain unramified 2-extensions
Akito Nomura
Illinois J. Math. 46(4): 1279-1286 (Winter 2002). DOI: 10.1215/ijm/1258138479


We study the inverse Galois problem with restricted ramification. Let $K$ be an algebraic number field and $G$ be a $2$-group. We consider the question whether there exists an unramified Galois extension $M/K$ with Galois group isomorphic to $G$. We study this question using the theory of embedding problems. Let $L/k$ be a Galois extension and $(\varepsilon): 1\to \mathbf{Z}/2\mathbf{Z}\to E\to \operatorname{Gal} (L/k)\to 1$ a central extension. We first investigate the existence of a Galois extension $M/L/k$ such that the Galois group $\operatorname{Gal} (M/k)$ is isomorphic to $E$ and any finite prime is unramified in $M/L$. As an application, we prove the existence of an unramified extension over cyclic quintic fields with Galois group isomorphic to $32{\Gamma}_5a_2$ under the condition that the class number is even. We also consider the Fontaine-Mazur-Boston Conjecture in the case of abelian $l$-extensions over $\mathbf{Q}$.


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Akito Nomura. "Notes on the existence of certain unramified 2-extensions." Illinois J. Math. 46 (4) 1279 - 1286, Winter 2002.


Published: Winter 2002
First available in Project Euclid: 13 November 2009

zbMATH: 1024.12005
MathSciNet: MR1988263
Digital Object Identifier: 10.1215/ijm/1258138479

Primary: 12F12
Secondary: 11R29, 11R32

Rights: Copyright © 2002 University of Illinois at Urbana-Champaign


Vol.46 • No. 4 • Winter 2002
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